On linear congruence relations for Kubota–Leopoldt 2-adic L-functions

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• 作者：Urbanowicz ; Jerzy ; van Wamelen ; Paul
• 关键词：11C20 ; 11R42 ; 11S40 ; 2-Adic L-functions ; Congruences for special values of L-functions ; Generalized Vandermonde determinants
• 刊名：Journal of Number Theory
• 出版年：2003
• 出版时间：January, 2003
• 年：2003
• 卷：98
• 期：1
• 页码：195-216
• 全文大小：209 K

文摘

Our purpose in the paper is to find the most general linear congruence relation of the Hardy–Williams type for linear combinations of special values of Kubota–Leopoldt 2-adic L-functions L₂(k,χω^1−k) with k running over any finite subset of Z not necessarily consisting of consecutive integers (see Acta Arith. 47 (1986) 263; Publ. Math. Fac. Sci. Besançon, Théorie des Nombres, 1995/1996; Publ. Math. Debrecen 56 (2000) 677 and cf. Mathematics and Its Applications, Vol. 511, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000). If k runs over finite subsets of Z consisting of consecutive integers see Compositio Math. 111 (1998) 289; Publ. Math. Debrecen 56 (2000) 677; Hardy and Williams, 1986; Compositio Math. 75 (1990) 271; Acta Arith. 71 (1995) 273; 52 (1989) 147; J. Number Theory 34 (1990) 362. In order to obtain the most general congruences of this type we make use of divisibility properties of the generalized Vandermonde determinants obtained in Spiez et al. (Divisibility properties of generalized Vandermonde and Cauchy determinants, Preprint 627, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2002). This allows us to simplify our main Theorem 2 and obtain Theorem 3 where the most general form of the linear congruence relation is given.